Cheryl Crane, daughter of Lana Turner, is writing a book about her life, to be published by Arbor House. Turner, according to Harry Haun in the New York Daily News, is cooperating with the project. Says Cliff Jahr, who is writing the book with Crane, "It won't be a 'Mommie Dearest' as much as it'll be a 'Little Girl Lost.' "

Crane made headlines in the fatal stabbing of mobster Johnny Stompanato at her mother's house in 1958 . . . .

Marathon runner Grete Waitz will be in town Monday and Tuesday to promote her new running video. Waitz, 1983 world champion and silver medalist in the '84 Olympics, will appear at a press conference Monday at the Sheraton Washington Hotel with rock singer Laura Branigan, who provides musical accompaniment to Waitz on the video. The runner's publicist says the tape "will try to compete with Jane Fonda's exercise video" . . .

Also in town Monday and Tuesday will be author Jackie Collins, actress Joan Collins' sister. Jackie Collins, who will be promoting her new novel, "Lucky," will appear on Channel 5's "Panorama" and sign autographs at Crown Books Monday . . .

Opera star Placido Domingo checked into the Watergate Hotel's Piano Suite Wednesday night. He will be performing at the Capital Centre this weekend . . .

At the 1985 Norwegian Film Festival awards ceremony Wednesday night, the newly released "Kiss of the Spider Woman," directed by Hector Babenco, was chosen as the best foreign film, and director Susan Seidelman's "Desperately Seeking Susan" received the "Spreader of Joy" award . . .